[1003.0299]
The local B-polarization of the CMB: a very sensitive probe of cosmic defects

We present a new and especially powerful signature of cosmic strings and
other topological or non-topological defects in the polarization of the cosmic
microwave background (CMB). We show that even if defects contribute 1% or less
in the CMB temperature anisotropy spectrum, their signature in the local
$\tilde{B}$-polarization correlation function at angular scales of tens of arc
minutes is much larger than that due to gravitational waves from inflation,
even if the latter contribute with a ratio as big as $r\simeq 0.1$ to the
temperature anisotropies. Proposed B-polarization experiments, with a good
sensitivity on arcminute scales, may either detect a contribution from
topological defects produced after inflation or place stringent limits on them.
Even Planck should be able to improve present constraints on defect models by
at least an order of magnitude, to the level of $\ep <10^{-7}$. A future
full-sky experiment like CMBpol, with polarization sensitivities of the order
of $1\mu$K-arcmin, will be able to constrain the defect parameter $\ep=Gv^2$ to
a few $\times10^{-9}$, depending on the defect model.

Topological defects can source scalar, vector and tensor modes in the early universe. The vector modes have power on small scales and can generate E and B polarization; the B signal can be quite distinctive, and used to constrain defect models with future data.

This paper appears to take some previous results for the B-mode power spectrum and multiply them by l^4, so e.g. in Fig 1 the power is very blue. Of course to be consistent you also have to multiply the noise and the any other spectrum of interest by l^4 as well, so you seem to gain nothing by doing this. Is there some point I have missed?

The paper also defines a 'local' scalar [tex]\tilde{B}[/tex] by taking two derivatives of the polarization tensor. However you gain nothing by doing this; with noisy or non-band-limited data you cannot calculate derivatives on a scale L without having data available over a scale L - the non-locality just hits you in a different form (see astro-ph/0305545 and refs).

The main point is that vector components of defects' contribution to CMB polarization anisotropies peak at scales smaller than those from inflation.

On the other hand, the ordinary E- and B-modes depend non-locally on the Stokes parameters, so they cannot be used to put constraints on causal sources like defects using the angular correlation function of E- and B-modes on small scales. That is the reason why Baumann and Zaldarriaga [0901.0958] suggested using instead the local modes. Those are the true causal modes, written in terms of derivatives of the Stokes parameters.

These local B-modes then have power spectra that are much bluer than the non-local ones, and hence enhance the small scale (high-l) end of the spectrum. It is by looking at the angular correlation functions at small separations (tens of arcmin) that one has a chance to measure the defect's contribution to the local B-modes, and distinguish it from the one of inflation.

Of course, the usual white noise power spectrum for polarization will also be modified by this [tex]\ell^4[/tex] factor, but by a suitable gaussian smoothing of the data (following Baumann&Zaldarriaga), we can indeed obtain large signal to noise ratios for binned data at small angular scales.

Baumann&Zaldarriaga looked at the model-independent signature of inflation at angles [tex]\theta>2[/tex] degrees. What we have realiazed is that, although model-dependent, the signal at angles [tex]\theta < 1[/tex] degrees can be much more significant. In fact, the feature at small angles is rather universal. The differences between defect models (and we considered four different ones) is just in the height and width of the first and second oscillations in the angular correlation functions (related to the heigth and position of the angular power spectrum). Therefore, with sufficient angular resolution one could not only detect defects (if they are there) but also differentiate between different models.

I think it is clear from the normal power spectra that the sourced vector mode B-polarization peaks at much smaller scales than the gravitational wave spectrum: mostly scales sub-horizon at recombination as opposed to tensor modes which decay on sub-horizon scales. I agree that with low enough noise this is an interesting signal (and has been calculated many times before), though it needs to be distinguished from other possible vector mode sources like magnetic fields.

I thought the point of the Baumann paper was to make a nice picture showing visually the structure of the correlations. The E and B modes contain exactly the same information as the tilde versions; in the same way the WMAP7 papers make some nice plots of the polarization-temperature correlation to visually show a physical effect, but these constrain the same information as the usual power spectra. In the Gaussian limit the usual E/B spectra contain all the information on the defect power spectrum.

Only Q and U can actually be measured locally on the sky (in one pixel you cannot calculate any spatial derivatives). The two-point Q/U correlations can be calculated from the usual E and B spectra.